I would like to change some variables in a integral and encountered to an issue. I create here 2 simple examples to describe my questions:
Exp 1. Suppose we want to change $(x,y)$ to $(u,v)$ such that: $x= u + v$ and $y= -u -2v$.
Using the Chain Rule: $$dx dy = (du + dv) (-du -2dv) = -2du dv -dv du = -2du dv + du dv = -du dv.$$
On the other hand, using the Jacobian determinant formula, we need the absolute value of Jacobian determinant, which is $|-1| = 1$. So: $$dx dy = 1. du dv = du dv \ne -du dv.$$ (Ref: http://www.math24.net/change-of-variables-in-double-integrals.html)
What was wrong here?
Exp 2. With one variable, we have $dx = x^'_u du$ - so no absolute value is needed. What is the main difference between the one variable case and the multi-variables case?
I really appreciate if anyone could give me good references for this. Thanks in advance!